Problem: Solve for $x$ : $ 6|x - 8| + 6 = -2|x - 8| + 10 $
Explanation: Add $ {2|x - 8|} $ to both sides: $ \begin{eqnarray} 6|x - 8| + 6 &=& -2|x - 8| + 10 \\ \\ { + 2|x - 8|} && { + 2|x - 8|} \\ \\ 8|x - 8| + 6 &=& 10 \end{eqnarray} $ Subtract ${6}$ from both sides: $ \begin{eqnarray} 8|x - 8| + 6 &=& 10 \\ \\ { - 6} &=& { - 6} \\ \\ 8|x - 8| &=& 4 \end{eqnarray} $ Divide both sides by ${8}$ $ \dfrac{8|x - 8|} {{8}} = \dfrac{4} {{8}} $ Simplify: $ |x - 8| = \dfrac{1}{2}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 8 = -\dfrac{1}{2} $ or $ x - 8 = \dfrac{1}{2} $ Solve for the solution where $x - 8$ is negative: $ x - 8 = -\dfrac{1}{2} $ Add ${8}$ to both sides: $ \begin{eqnarray} x - 8 &=& -\dfrac{1}{2} \\ \\ {+ 8} && {+ 8} \\ \\ x &=& -\dfrac{1}{2} + 8 \end{eqnarray} $ Change the ${ + 8}$ to an equivalent fraction with a denominator of $2$ $ x = - \dfrac{1}{2} {+ \dfrac{16}{2}} $ $ x = \dfrac{15}{2} $ Then calculate the solution where $x - 8$ is positive: $ x - 8 = \dfrac{1}{2} $ Add ${8}$ to both sides: $ \begin{eqnarray} x - 8 &=& \dfrac{1}{2} \\ \\ {+ 8} && {+ 8} \\ \\ x &=& \dfrac{1}{2} + 8 \end{eqnarray} $ Change the ${ + 8}$ to an equivalent fraction with a denominator of $2$ $ x = \dfrac{1}{2} {+ \dfrac{16}{2}} $ $ x = \dfrac{17}{2} $ Thus, the correct answer is $x = \dfrac{15}{2} $ or $x = \dfrac{17}{2} $.